10 SPACES WITH OPERATORS. ORBIT SPACES 45
(metrizable) topology induced on G by that of ^R(I) is compatible with the group
structure of G, and that right Cauchy sequences in G are the same as Cauchy sequences
(in G) with respect to the norm on ^R(I) (7.1). Give an example of a right Cauchy
sequence in G which is not a left Cauchy sequence.
9. (a) Let G' be a complete metrizable group and let G0 ^ G' be a dense subgroup of
G'. Let G be the topplogical group obtained by giving G0 the discrete topology. The
identity mapping G -> G0 is a continuous bijective homomorphism, but its extension
by continuity to a homomorphism G -> G' (1 2.9.4) is not surjective.
(b) Let G be the dense subgroup Q2 of R2, let 6 be an irrational number, and let u
be the continuous homomorphism (x, y)\- >x + By of G into R. Let G' = w(G). The
mapping #, considered as a homomorphism of G into G', is bijective, but its continuous
extension to a homomorphism of R2 into R is not injective.
(c) Use (a) and (b) to construct an example of two complete groups Gj, G2 and a
continuous bijective homomorphism 'u of a dense subgroup H! of Gx onto a dense
subgroup H2 of G2 , such that the continuous extension of u to a homomorphism of
Gi into G2 is neither injective nor surjective.
10. Let / be a continuous homomorphism of a subgroup H ^ {0} of R into a locally
compact metrizable group G. Show that if /is not an isomorphism of H onto the
subgroup /(H) of G, then /(H) is relatively compact in G. (Reduce to the case where
H is closed in R and/(H) is dense in G. Begin by showing that, for each neighborhood
W of the neutral element e in G, the set/-1(W) is unbounded, and deduce that
/(H n R+) is dense in G. If V is a compact symmetric neighborhood of e in G, show
that there exist a finite number of elements tt > 0 in H such that the neighborhoods
/(/,)V cover V. For each x e G, let A* be the set of all t e H such that /(/) e xV.
Show that, if t e A* , there exists a tt such that / — /, e Ax , and deduce that, if I is the
largest compact interval with origin 0 in R containing all the // , then I n Ax is not
empty. Deduce that G C/(I) • V is compact.)
11. Let G be a commutative metrizable topological group, and let d be an invariant
distance defining the topology of G. Let h be the HausdorrT distance on 3f(G) cor-
responding to d (Section 3.16, Problem 3). Show that if M, N, P, Q are four bounded
closed subsets of G, then /z(MP, NQ) <J h(M, N) + A(P, Q).
10. SPACES WITH OPERATORS. ORBIT SPACES
Let G be a group and E a set. An action (or left action) of G on E is a
mapping (s, x)t->s - x of G x E into E satisfying the following conditions:
(1) If e is the neutral element of G, then e - x = x for all x e E.
(2) For all s, t in G we have s - (t - x) = (st) - x for all x e E.
These conditions imply that s"1 • (s • x) = x for all s e G and all xeE;
hence for each s e G the mapping x\-*s • x is a bijection of E onto E, and the
inverse bijection is x\-*s~l • x.
For each x e E, the set G • x of elements s • x where s e G is called the
orbit of x (for the given action of G on E). The set S* of elements s e G suchem